This seems pretty obviously like a bad idea to me. What's to stop someone from erecting a large structure inside a public park, stealing the right to use that land from the public and taking it for themselves?

Having lived in Seattle until very recently, I can attest first hand that the answer to that question is: nothing. So your assertion here seems like it would lead to a state where the first person to create a structure on public land and live in it effectively converts that land into their own private property. To be frank, that is absurd.